A model for cell-cell adhesion, based on a model originally proposed by
N. J. Armstrong, K. J. Painter, and J. A. Sherratt (2006), is studied.
The model consists of a non-linear partial differential equation for the
cell density in an N-dimensional infinite domain. It has a nonlocal flux
term which models the component of cell motion attributable to cells
having formed bonds with other cells within its sensing radius. Using
the theory of fractional powers of analytic semigroup generators and
working in spaces with bounded uniformly continuous derivatives, the
local existence of classical solutions is proved. Positivity and
boundedness of solutions is then established, leading to global
existence of solutions. Finally, the asymptotic behaviour of solutions
about the the spatially uniform state is considered. The model is
illustrated by simulations that can be applied to in vitro wound closure
experiments.

This is joint work with S. A. Gourley, R. Villella-Bressan and G. F.
Webb.

Contact person:
Glenn Webb

Reed Muller codes are old. They were among the very first algebraic
error correcting codes to be discovered and analyzed and they find
application today as spreading sequences in spread spectrum wireless
communication. Compressed sensing is much more modern. The idea of
capturing attributes of a signal with very few measurements has wide
applicability and we will describe how second order Reed-Muller codes
lead to a new deterministic framework.

Contact person:
Alex Powell

From every countable group G or measure preserving
action of G on a probability space X, one can construct a von Neumann
algebra. A central theme in the theory of von Neumann algebras is
understading how much of the group or group action is “remembered”
by its von Neumann algebra. In this talk, I will survey recent results
which provide the first classes of groups and group actions that can be
completely recovered from their von Neumann algebras.

Contact person:
Dietmar Bisch

With every relational structure (e.g. a directed graph) one can
associate the set of operations preserving the relations of this
structure (the edge-relation in case of a digraph). This correspondence
is at the core of a connection between computational complexity of
non-uniform constraint satisfaction problems and universal algebra. In
the talk I will briefly outline the connection, introduce the most
significant advances and discuss the impact on both areas.

Contact person:
Ralph McKenzie

Finding good partitions of graphs and hypergraphs is important to many
combinatorial problems and has applications to other fields such as
VLSI designs. Graph and hypergraph partitions have been been studied
extensively by researchers from various fields using various techniques.
In this talk I will discuss a number of such problems. I will also
present some recent results we obtained using structural and
probabilistic methods.

Contact person:
Mark Ellingham

In 1975 Szemerédi proved the remarkable combinatorial fact
that any subset of the integers having positive upper density contains
arbitrarily long arithmetic progressions. Although Szemerédi’s proof
was purely combinatorial, shortly afterwards Furstenberg gave a new
proof of Szemerédi’s Theorem using a conversion to an assertion of
`multiple recurrence’ for probability-preserving systems, which he
then proved using newly-developed machinery in ergodic theory.

Furstenberg’s proof gave rise to a new subdiscipline called `Ergodic
Ramsey Theory’, which went on to provide proofs for several other
extremal results in different combinatorial settings. More recent
work has provided a much more detailed picture of the structures that
underlie these ergodic theoretic analyses, and offered a clearer
insight into the connections between this field and purely
combinatorial approaches to the same results. In this talk I will
describe some of this interplay and sketch how it has led to both new
advances within ergodic theory and to a new approach to the
multidimensional generalizations of multiple recurrence and
Szemerédi’s Theorem.

Contact person:
Guoliang Yu

A classical spin network is a cubic (i.e. 3-regular) graph whose edges
are colored by natural numbers. Its evaluation is an integer. When we
scale the colors of all edges by the same amount, we get a sequence of
integers, whose asymptotics captures important information about the
graph. We will discuss (a) the existence of asymptotic expansions, using
the theory of G-functions and algebraic geometry, (b) the computation of
asymptotic expansions using the Zeilberger-Wilf theory of holonomic
sequences, (d) arithmetic invariants of spin networks and (d) their
application to combinatorics and low dimensional topology. We will
present numerous examples to illustrate the abstract/concrete principles
involved. This is joint work with Roland van der Veen.

Contact person:
Dietmar Bisch

In this survey talk I will explain the basic ideas of the theory of
abstract evolution equations as well as present applications to partial
differential equations. I intend to show how the concept of maximal
regularity naturally comes into play for quasilinear parabolic problems.
An outline of its impact on the analysis of free boundary problems will
be given.

Contact person:
Gieri Simonett

An Einstein metric is by definition a Riemannian metric of constant
Ricci curvature. One would like to completely determine which smooth
compact n-manifolds admit such metrics. In this talk, I will describe
recent progress regarding the 4-dimensional case. These results
specifically concern 4-manifolds that also happen to carry either a
complex structure or a symplectic structure.

Contact person:
Ioana Suvaina

A common theme in modern mathematics is to pretend that an arbitrary
ring is the ring of functions on some space, even if the ring is
non-commutative or nilpotent and thus not the ring of functions on an
honest space. Similarly, it’s fruitful to look at categories which look
like the representation theory of a finite group even if they don’t come
from an honest group. Such a category is called a fusion category, and
fusion categories can be thought of as the representation theory of a
“finite quantum group.” I’ll begin by motivating the definition of a
fusion category and giving several examples of fusion categories.
The bulk of the talk will be spent justifying the study of fusion
categories by explaining how they’re related to other subjects in
algebra, topology, and operator algebras. Towards the end of the talk
I’ll discuss current research in the field, with examples drawn both
from my own work and the work of others.

Contact person:
Dietmar Bisch

A Riemannian manifold with boundary is said to be boundary rigid
if its metric is uniquely determined by the boundary distance function,
that is the restriction of the distance function to the boundary.
Loosely speaking, this means that the Riemannian metric can be recovered
from measuring distances between boundary points only. The goal is to
show that certain classes of metrics are boundary rigid (and, ideally,
to suggest a procedure for recovering the metric).

To visualize that, imagine that one wants to find out what the Earth is
made of. More generally, one wants to find out what is inside a solid
body made of different materials (in other words, properties of the
medium change from point to point). The speed of sound depends on the
material. One can “tap” at some points of the surface of the body and
“listen when the sound gets to other points”. The question is if this
information is enough to determine what is inside.

This problem has been extensively studied from the PDE viewpoint: the
distance between boundary points can be interpreted as a “travel time”
for a solution of the wave equation. Hence this becomes a classic
Inverse Problem when we have some information about solutions of a
certain PDE and want to recover its coefficients. For instance such
problems naturally arise in geophysics (when we want to find out what is
inside the Earth by sending sound waves), medical imaging etc.

In a joint project with S. Ivanov we suggest an alternative geometric
approach to this problem. In our earlier work, using this approach we
were able to show boundary rigidity for metrics close to flat ones (in
all dimensions), thus giving the first class of boundary rigid metrics
of non-constant curvature beyond two dimensions. We are now able to
extend this result to include metrics close to a hyperbolic one.

The approach grew out of another long-term project of studying
surface area functionals in normed spaces, which we have been working on
for more than ten years. There are a number of related issues
regarding area-minimizing surfaces in Riemannian manifolds. The talk
gives a non-technical survey of ideas involved. It assumes no background
in inverse problems and is supposed to be accessible to a general math
audience (in other words, we will not get into any technical details of
the proofs).

Contact person:
Mark Sapir

Mechanism design theory is concerned with social decision making when
information is privately held by n individuals. Individual
i‘s private information is described by his type
tⁱ. We consider direct mechanisms that assign an
outcome and an n-vector of individual payments to each
n-tuple of reported individual types. The allocation function
assigning outcomes to types is dominant strategy implementable (DISC) if
there is a payment function such that nobody ever has an incentive to
falsely report his type.

When there are a finite number m of outcomes, for a given
individual i and type vector of the other individuals, we can
equivalently describe i‘s type
tⁱ
by a vector
v^ti
in R^m, where
v_j^ti
is the value of the jth outcome to i when i
is of type tⁱ. This set of valuation types is used
to define a directed graph (the valuation graph) whose nodes are the
set of possible outcomes.

The Rockafellar-Rochet Theorem provides necessary and sufficient
conditions for an allocation function to be DISC in terms of the length
of cycles with an arbitrary number of directed arcs in the valuation
graph. The Saks-Yu Theorem shows that it is sufficient to only
consider 2-cycles if the set of possible valuation vectors is convex. We
show that some stronger implications follow if the set of valuations is
a convex product set. We also identify and exploit some geometric
properties of this problem.

This is joint work with Katherine Cuff, Sunghoon Hong, Jesse A.
Schwartz and Quan Wen.

Contact person:
Paul Edelman

I will give a survey of history and motivation for the study of
rigidity of higher rank group actions.
I will then focus on some recent work with Kalinin and Spatzier on
rigidity of Anosov actions of higher rank
abelian groups on tori and nilmanifolds.

Contact person:
Guoliang Yu

In 1890 G. Peano introduced a continuous function that maps the unit
interval onto the unit square. It is the first construction of a
space-filling curve. In 1891 Hilbert discovered another space-filling
curve by employing the technique of self-similarity. Later space-filling
curves have been constructed by many authors, all of which have the
properties that they are Hölder continuous and are measure
preserving. A natural question is whether a Peano curve exists for any
connected self-similar set such as the Sierpinski Gasket . We discuss
this question in this talk. We also discuss some interesting
applications of such Peano curves.

Contact person:
Akram Aldroubi